Mach Like Principle from Conserved Charges

Abstract

We study models where the gauge coupling constants, masses and the gravitational constant are functions of some conserved charge in the universe, and furthermore a cosmological constant that depends on the total charge of the universe. We first consider the standard Dirac action, but where the mass and the electromagnetic coupling constant are a function of the charge in the universe and afterwards extend this to curved spacetime and consider gauge coupling constants, the gravitational constant and the mass as a function of the charge of the universe, which represent a sort of Mach principle for all the constants of nature. In the flat space formulation, the formalism is not manifestly Lorentz invariant, however Lorentz invariance can be restored by performing a phase transformation of the Dirac field. One interesting model of this type is one where the action is invariant under rescalings of the Dirac wave function. In the curved space time formulation, there is the additional feature that some of the equations of motion break the general coordinate invariance also, but in a way that can be understood as a coordinate choice only, so the equations are still of the General Relativity type, but with a certain natural coordinate choice, where there is no current of the charge. We have generalized what we have done and also constructed a cosmological constant which depends on the total charge of the universe. We discuss how these ideas work when the space where the charges live is finite. If we were to use some only approximately conserved charge for these constructions, like say baryon number (in the context of the standard model), this will lead to corresponding violations of Lorentz symmetry in the early universe for example. We also briefly discuss another non-local formulations where the coupling constants are functions of the Pontryagin index of some non-abelian gauge field configurations. The construction of charge dependent contributions can also be motivated from the structure of the “infra-red counter terms” needed to cancel infra red divergences for example in three dimensions.

Appendices

Appendix A: The Lorentz Invariance of Q from the Conservation of the 4 Current

We will define the four Dirac current to be:

$$ j^{\mu}=\bar{\psi}(y)\gamma^{\mu}\psi(y) $$

(88)

and since we know after we use the equation of motion that the four dimension current is conserved:

$$ \partial_{\mu}j^{\mu}=0 $$

(89)

the total charge is:

$$ Q=\int{j^{0}(x)\,d^{3}x} $$

(90)

Following Weinberg [10, 11] we can rewrite the charge as:

$$ Q=\int{d^{4}x\,j^{\mu}\partial_{\mu}\theta \bigl(n^{\nu}x_{\nu}\bigr)} $$

(91)

where θ(a) is a step function, and by definition n _i =0, n ₀=1. If we perform a Lorentz transformation on Q then:

(92)

where $n_{\lambda}x'^{\lambda}=n'_{\rho}x^{\rho}$ so

$$ Q'=\int{d^{4}x\,j^{\mu}\partial_{\mu} \theta\bigl(n'^{\nu}x_{\nu}\bigr)} $$

(93)

if we using Eq. (89) then we can write:

$$ Q'-Q=\int{d^{4}x\,\partial_{\mu} \bigl[j^{\mu}\bigl\lbrace\theta\bigl(n'_{\nu}x^{\nu } \bigr)-\theta\bigl(n_{\nu}x^{\nu}\bigr) \bigr\rbrace\bigr]} $$

(94)

The current j ^μ can be presumed to vanish if |x|⟶∞ with t fixed, whereas the function $\theta(n'_{\nu}x^{\nu})-\theta(n_{\nu}x^{\nu})$ vanishes if |t|⟶∞ with x fixed, so Q′−Q=0 so the charge is invariant under Lorentz transform because n _ν and $n'_{\nu}$ is time like.

Appendix B: The Klein Gordon and Schroedinger Cases

2.1 B.1 The Klein Gordon Case

Until now we have steadied the Dirac field, now we will show how these affects look in the Klein Gordon case.

2.2 B.2 When the Mass is a Function of All the Charge in the Universe on the Klein Gordon Field

We will begin with the action of Klein Gordon equation:

$$ S=\int{d^{4}x\,\bigl[\bigl(i\partial_{\mu}\phi^{*} +eA_{\mu}\phi^{*}\bigr) \bigl(-i\partial^{\mu} \phi+eA^{\mu}\phi\bigr)-m^{2}\phi^{*}\phi\bigr]} $$

(95)

where we will take:

(96)

which is the total charge in the universe by the definition of Klein Gordon field. So by variation we will get:

(97)

from this we get the equation of motion:

(98)

If we do the transformation

$$ A^{0}\longrightarrow A^{0}+ \frac{i\lambda_{1}b}{e}\delta\bigl(y^{0}-t_{0}\bigr) $$

(99)

and

$$ \phi=e^{\lambda_{2} b\theta(y^{0}-t_{0})}\phi_{0} $$

(100)

where b=iλ∫ϕ ^∗ ϕ  d ⁴ y then we have that

(101)

(102)

So, if we use Eq. (99) in Eq. (98) than we will get:

(103)

and finally we put Eq. (100) and we get:

(104)

If we need that Eq. (104) will be like ordinary Klein Gordon equation we need that:

(105)

(106)

for which the solutions are $\lambda_{2}=\frac{1}{\sqrt{2}} ,\, -\frac{1}{{\sqrt{2}}}$ and $\lambda_{1}=1-\frac{1}{\sqrt{2}} ,\, 1+\frac{1}{\sqrt{2}}$ respectively, where for these solutions Eq. (104) will become:

$$ \bigl[(i\partial_{\mu}-eA_{\mu}) \bigl(i\partial^{\mu}-eA^{\mu} \bigr)-m^{2}\bigr]\phi_{0}=0 $$

(107)

2.3 B.3 Coupling Constant Depending of Charge in Klein Gordon

We will begin with the action of Klein Gordon equation:

$$ S=\int{d^{4}x\,\bigl[\bigl(i\partial_{\mu}\phi^{*} +eA_{\mu}\phi^{*}\bigr) \bigl(-i\partial^{\mu} \phi+eA^{\mu}\phi\bigr)-m^{2}\phi^{*}\phi\bigr]} $$

(108)

where we will take:

(109)

which is the total charge in the universe by the definition of Klein Gordon. So by variation we will get:

(110)

where we know that $\int{d^{4}x\,A_{\mu}(\phi^{*}i\stackrel {\leftrightarrow}{\partial^{\mu}}\phi-2eA^{\mu}\phi^{*}\phi)}=\int {d^{4}x\,A_{\mu}j^{\mu}}$ so the equation of motion will become:

(111)

The solution of the equation is the same as in the last section but with b=iλ∫d ⁴ x  A _μ j ^μ.

2.4 B.4 The Schroedinger Case

We can see that also in Schrodinger equation case these effects can be studied. The action for the Schrodinger equation is [8]:

(112)

we can take the coupling constant as “e” and “m” and follow the same approach as before where the coupling constant is proportional to the total charge in the universe, for example [16]:

$$ e(Q)=\lambda_{1}\int{\psi^{*}\psi\,d^{3}x} $$

(113)

and

$$ m(Q)=\lambda_{2}\int{\psi^{*}\psi\,d^{3}x} $$

(114)

where

$$ Q=\int{\psi^{*}\psi\,d^{3}x}=\int{\rho\,d^{3}x} $$

(115)

The equation of motion will be like the standard Schrodinger equation, if we will preform the transformation:

$$ \psi=e^{i(b_{e}+b_{m})\theta(t-t^{0})}\psi_{0} $$

(116)

where ψ ₀ is the solution of the standard Schrodinger equation, and “b’s” are

$$ b_{e}=-\lambda_{1}\int{ \biggl\lbrace\frac{1}{2m} \bigl[i\bigl(\nabla\psi^{*}\bigr)\mathbf {A}\psi-i\psi^{*} \mathbf{A}(\nabla\psi)+2e\mathbf{A}^{2}\psi^{*}\psi \bigr]- \psi^{*}\psi V\biggr\rbrace\,d^{4}x} $$

(117)

and

$$ b_{m}=\frac{1}{2m^{2}}\lambda_{2}\int{ \bigl(\nabla \psi^{*}-i\psi^{*}e\mathbf {A}\bigr) (\nabla\psi+ie \mathbf{A}\psi)\,d^{4}x} $$

(118)